LO 66.9: Describe dispersion, explain its causes, and describe methods for

LO 66.9: Describe dispersion, explain its causes, and describe methods for controlling forms of dispersion.
For portfolio managers, dispersion refers to the variability of returns across client portfolios. One dispersion measure is the difference between the maximum return and minimum return over a period for separately managed client accounts.
Managers can reduce dispersion by reducing differences in asset holdings between portfolios and differences in portfolio betas though better supervision and control. Other causes of dispersion are outside the managers control. Different portfolio constraints for different accounts will unavoidably increase dispersion (e.g., not being able to invest in derivatives or other asset classes).
Of course, if all client accounts were identical there would be no dispersion. All accounts will not be identical in the presence of transaction costs, however. The existence of transaction costs implies that there is some optimal level of dispersion. To understand the tradeoff between transaction costs and dispersion, consider a managed portfolio that is currently 60% stocks and 40% bonds. The manager knows the optimal portfolio is 62% stocks and 38% bonds, but transaction costs from rebalancing would reduce returns more than the change to optimal weights would increase them.
If the manager acquires a second client, he can set portfolio weights to 62% and 38% for that clients account. Because one client has a 60/40 portfolio and the other has a 62/38 portfolio, there will be dispersion. Clearly, higher transaction costs lead to greater dispersion. If the manager eliminates dispersion by matching the new client portfolio to the existing client portfolio, returns from the new information will be sacrificed. If the manager eliminates dispersion by rebalancing the existing client portfolio, the transaction costs of this rebalancing will reduce overall portfolio return. Given transaction costs, there is an optimal level of dispersion that balances transaction costs and gains from rebalancing.
A greater number of portfolios and higher active risk will both increase optimal dispersion, and for a given number of portfolios, dispersion is proportional to active risk. As long as alphas and risk are not constant (an unlikely occurrence) dispersion will decrease over time and eventually convergence (of account returns) will occur. Flowever, there is no certainty as to the rate at which it will occur.
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Topic 66 Cross Reference to GARP Assigned Reading – Grinold and Kahn, Chapter 14
Ke y Co n c e pt s
LO 66.1 The inputs into the portfolio construction process are the current portfolio, the alphas, covariance estimates, transaction costs, and active risk aversion. Except for the current portfolio, these inputs are all subject to estimation error and possible bias.
LO 66.2 Refining alphas is an alternative to including constraints (e.g., no short selling or maximum deviations from benchmark weights) in the portfolio optimization process. Using refined alphas and then performing optimization can achieve the same goal as a constrained optimization approach, but has the advantage of focusing on the alpha inputs and the effects of individual constraints on portfolio returns.
LO 66.3 Neutralization can remove undesirable portfolio risks. Benchmark neutralization can reduce active risk by matching active portfolio beta to that of the benchmark portfolio. Cash neutralization eliminates any active cash position in the portfolio. Risk-factor neutralization matches specific factor risks in the active portfolio to those of the benchmark.
LO 66.4 Transaction costs have several implications. First, they may make it optimal not to rebalance even with the arrival of new information. Second, transaction costs increase the importance of robust alpha estimates. The fact that transaction costs occur at a point in time while the benefits of the portfolio adjustments occur over the investment horizon complicates analysis and makes rebalancing decisions dependent on the estimated holding period of portfolio assets.
LO 66.3 Practical issues in portfolio construction include determining the level of risk aversion, the optimal risk, and the alpha coverage. The inputs in computing the level of risk aversion must be accurate. Including the aversion to specific risk factors can help a manager address the risks of a position with a large potential loss and the dispersion across separately managed portfolios. Proper alpha coverage addresses situations in which the manager has alpha estimates for stocks that have zero weight in (are not included in) the benchmark or does not have alpha estimates for some stocks in the benchmark portfolio.
LO 66.6 In the process of portfolio revisions and rebalancing, there are tradeoffs between alpha, risk, transaction costs, and the investment horizon. The manager may choose to be conservative, given the uncertainty regarding these inputs. Also, the shorter the horizon, the more uncertain the alpha, which means the manager should choose an optimal time horizon where the certainty of the alpha is sufficient to justify a trade given the transaction costs.
2018 Kaplan, Inc.
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Topic 66 Cross Reference to GARP Assigned Reading – Grinold and Kahn, Chapter 14
LO 66.7 Because of transaction costs, there will be an optimal no-trade region when new information arrives concerning the alpha of an asset, the costs of rebalancing the portfolio outweigh the expected incremental returns. That region is determined by the level of risk aversion, a portfolios active risk, the marginal contribution of rebalancing to active risk, and transaction costs.
LO 66.8 A screen may be as simple as screening for assets with the highest estimated alphas or as a method of assigning relative ranks based on estimated alphas.
Stratification applies screening separately to categories of stocks and weights the active portfolio across these categories with their weights in the benchmark portfolio.
Linear programming is an improvement on stratification in that the optimal portfolio is structured to closely resemble the benchmark with respect to such characteristics (risk factors) as industry groups, firm size, volatility, and beta.
Quadratic programming improves on the linear programming methodology by explicitly considering alpha, risk, and transaction costs. It is theoretically the best optimization method, incorporating the most information; however, the value added in the active portfolio is quite sensitive to the level of estimation error in the covariance inputs.
LO 66.9 For a manager with separately managed accounts, dispersion of client returns will result when the portfolios are not identical. The basic causes of dispersion are the different histories and cash flows of each of the clients. A manager can control this source of dispersion by trying to increase the proportion of assets that are common to all portfolios.
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Topic 66 Cross Reference to GARP Assigned Reading – Grinold and Kahn, Chapter 14
Co n c e pt Ch e c k e r s
The most measurable of the inputs into the portfolio construction process is(are): A. B. C. D. the active risk aversion.
the position alphas. the transaction costs. the current portfolio.
Which of the following is correct with respect to adjusting the optimal portfolio for portfolio constraints? A. No reliable method exists. B. By refining the alphas and then optimizing, it is possible to include constraints
of both the investor and the manager.
C. By refining the alphas and then optimizing, it is possible to include constraints
D. By optimizing and then refining the alphas, it is possible to include constraints
of the investor, but not the manager.
of both the investor and the manager.
An increase in which of the following factors will increase the no-trade region for the alpha of an asset? I. Risk aversion. II. Marginal contribution to active risk. A. I only. B. II only. C. Both I and II. D. Neither I nor II.
Which of the following statements most correctly describes a consideration that complicates the incorporation of transaction costs into the portfolio construction process? A. The transaction costs and the benefits always occur in two distinct time periods. B. The transaction costs are uncertain while the benefits are relatively certain. C. There are no complicating factors from the introduction of transaction costs. D. The transaction costs must be amortized over the horizon of the benefit from
the trade.
A manager has forecasts of stocks A, B, and C, but not of stocks D and E. Stocks A, B, and D are in the benchmark portfolio. Stocks C and E are not in the benchmark portfolio. Which of the following is correct concerning specific weights the manager should assign in tracking the benchmark portfolio? A. Wq = 0. B. wD = 0. C. wc = (wA + wB)/2. D. wc = wD = wE.
2018 Kaplan, Inc.
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Topic 66 Cross Reference to GARP Assigned Reading – Grinold and Kahn, Chapter 14
Co n c e pt Ch e c k e r An s w e r s 1. c 2. B
The current portfolio is the only input that is directly observable.
The approach of first refining alphas and then optimizing can replace even the most sophisticated portfolio construction process. With this technique, both the investor and manager constraints are considered.
3. C This is evident from the definition of the no-trade region for the alpha of the asset. alpha of asset < [2 x (risk aversion) x (active risk) x (marginal contribution to active risk)] + [2 x (risk aversion) x (active risk) x (marginal contribution to active risk)] - (cost of selling) < alpha of asset < [2 x (risk aversion) x (active risk) x (marginal contribution to active risk)] + (cost of purchase) 4. D A challenge is to correctly assign the transaction costs to projected future benefits. The transaction costs must be amortized over the horizon of the benefit from the trade. The benefits (e.g., the increase in alpha) occur over time while the transaction costs generally occur at a specific time when the portfolio is adjusted. 5. A The manager should assign a tracking portfolio weight equal to zero for stocks for which there is a forecast but that are not in the benchmark. A weight should be assigned to stock D, and it should be a function of the alphas of the other assets. Page 72 2018 Kaplan, Inc. The following is a review of the Risk Management and Investment Management principles designed to address the learning objectives set forth by GARP. This topic is also covered in: Po r t f o l i o Ri s k : A n a l y t ic a l M e t h o d s Topic 67 Ex a m Fo c u s Due to diversification, the value at risk (VaR) of a portfolio will be less than or equal to the sum of the VaRs of the positions in the portfolio. If all positions are perfectly correlated, then the portfolio VaR equals the sum of the individual VaRs. A manager can make optimal adjustments to the risk of a portfolio with such measures as marginal VaR, incremental VaR, and component VaR. This topic is highly quantitative. Be able to find the optimal portfolio using the excess-return-to-marginal VaR ratios. For the exam, understand how correlations impact the measure of portfolio VaR. Also, it is important that you know how to compute incremental VaR and component VaR using the marginal VaR measure. We have included several examples to help with application of these concepts. Portfolio theory depends a lot on statistical assumptions. In finance, researchers and analysts often assume returns are normally distributed. Such an assumption allows us to express relationships in concise expressions such as beta. Actually, beta and other convenient concepts can apply if returns follow an elliptical distribution, which is a broader class of distributions that includes the normal distribution. In what follows, we will assume returns follow an elliptical distribution unless otherwise stated.

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